In this paper we introduce a generalization of Picard groups to derived categories of algebras. First we study general properties of this group. Then we consider easy particular algebras such as commutative algebras, where we reduce to the classical case. Finally, we define and study a homomorphism of the braid group to the Picard group of the derived category of a Brauer tree algebra. In the smallest case we show that this homomorphism is injective and that its image is of finite index.

We study conjugacy invariants for completely positive maps that are inspired by the concept of curvature introduced for commuting d-tuples of contractions by Arveson.

The concept of generalized functions taking values in a differentiable manifold is extended to a functorial theory. We establish several characterization results which allow a global intrinsic formulation both of the theory of manifold-valued generalized functions and of generalized vector bundle homomorphisms. As a consequence, a characterization of equivalence that does not resort to derivatives (analogous to scalar-valued cases of Colombeau's construction) is achieved. These results are employed to derive a point value description of all types of generalized functions valued in manifolds and to show that composition can be carried out unrestrictedly. Finally, a new concept of association adapted to the present setting is introduced.

In this note we consider a stability property of p-harmonic functions in general metric measure spaces. Despite the fact that there may not be a corresponding differential equation, it is shown here that a family of uniformly convergent p-harmonic functions converges to a p-harmonic function; that is, the collection of p-harmonic functions forms a closed subspace in a certain Ho¨lder class. As a consequence, it is shown that a family of p-harmonic functions bounded in a Sobolev-type space (called the Newtonian space) has a sequence that converges locally uniformly in the domain of harmonicity to a p-harmonic function. This result is used to construct p-harmonic functions on unbounded domains. We also use this convergence result to prove a characterization of a parabolicity property of metric measure spaces. This characterization has been given for Riemannian manifolds by Holopainen, and the result here is a generalization of Holopainen's result.

In this paper we give lower bounds for the Mislin genus of the symplectic groups $\mathrm{Sp}(m)$. This result appears to be the exact analogue of Zabrodsky's theorem concerning the special unitary groups $\mathrm{SU}(n)$ . It is achieved by the determination of the stable genus of the quasi-projective quaternionic spaces $Q\mathbb{H}(m)$ , following the approach of McGibbon. It leads to a symplectic version of Zabrodsky's conjecture, saying that these lower bounds are in fact the exact cardinality of the genus sets. The genus of $\mathrm{Sp}(2)$ is well known to contain exactly two elements. We show that the genus of $\mathrm{Sp}(3)$ has exactly 32 elements and see that the conjecture is true in these two cases.

Independently, we also show that any homotopy type in the genus of $\mathrm{Sp}(m)$ fibers over the sphere $S^{4m-1}$ with fiber in the genus of $\mathrm{Sp}(m-1)$ , and that any homotopy type in the genus of $\mathrm{SU}(n)$ fibers over the sphere $S^{2n-1}$ with fiber in the genus of $\mathrm{SU}(n-1)$ . Moreover, these fibrations are principal with respect to some appropriate loop structures on the fibers. These constructions permit us to produce particular spaces realizing the lower bounds obtained.

The aim of this paper is to provide an analysis of non-linear approximation in the $L_p$-norm $p = d / (d - 1)$ of functions of bounded variation on $\mathbb{R}^d$ with $d > 1$ by polynomials in the Haar system. The exponent $p$ is the natural exponent as it is the correct exponent in the Sobolev inequality. The approximation schemes that we discuss in this paper are mostly related to Haar thresholding and $m$-term approximation. These problems for $d = 2$ are studied in detail in a paper by Cohen, DeVore, Petrushev and Xu. The main aim of this paper is to extend their results to the case $d \geq 2$.

We obtain the optimal order of the $m$-term Haar approximation and prove the stability of Haar thresholding in the BV-norm. As one of the main tools, we establish the boundedness of certain averaging projections in BV.

The requirement that a (non-Einstein) Kähler metric in any given complex dimension $m > 2$ be almost-everywhere conformally Einstein turns out to be much more restrictive, even locally, than in the case of complex surfaces. The local biholomorphic-isometry types of such metrics depend, for each $m > 2$, on three real parameters along with an arbitrary Kähler–Einstein metric $h$ in complex dimension $m - 1$. We provide an explicit description of all these local-isometry types, for any given $h$. This result is derived from a more general local classification theorem for metrics admitting functions that we call special Kähler–Ricci potentials.

Our results are of three types. First, we describe a general procedure of adjoining polynomial variables to $A_\infty$-ring spectra whose coefficient rings satisfy certain restrictions. A host of examples of such spectra is provided by killing a regular ideal in the coefficient ring of $MU$, the complex cobordism spectrum. Second, we show that the algebraic procedure of adjoining roots of unity carries over in the topological context for such spectra. Third, we use the developed technology to compute the homotopy types of spaces of strictly multiplicative maps between suitable $K(n)$-localizations of such spectra. This generalizes the famous Hopkins–Miller theorem and gives strengthened versions of various splitting theorems.

Let $K$ be a knot with an unknotting tunnel $\gamma$ and suppose that $K$ is not a $2$-bridge knot. There is an invariant $\rho = p/q \in \mathbb{Q}/2 \mathbb{Z}$, with $p$ odd, defined for the pair $(K, \gamma)$.

The invariant $\rho$ has interesting geometric properties. It is often straightforward to calculate; for example, for $K$ a torus knot and $\gamma$ an annulus-spanning arc, $\rho(K, \gamma) = 1$. Although $\rho$ is defined abstractly, it is naturally revealed when $K \cup \gamma$ is put in thin position. If $\rho \neq 1$ then there is a minimal-genus Seifert surface $F$ for $K$ such that the tunnel $\gamma$ can be slid and isotoped to lie on $F$. One consequence is that if $\rho(K, \gamma) \neq 1$ then $\mathrm{genus}(K) > 1$. This confirms a conjecture of Goda and Teragaito for pairs $(K, \gamma)$ with $\rho(K, \gamma) \neq 1$.